Ayudantia 3

Vista previa del material en texto

Macroeconomía II – Problem Set
TA Session 3
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Some of those exercises will be solved on the ayudantia of September 7, 2018.
Money demand
Exercise 1
Consider an economy in which there is a single good that can be used as capital or for
consumption. Time is discrete and indexed by t ∈ {−1, 0, 1, 2, ...}. Consider the following
problem of the representative household:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t. Ptct + Ptkt +Bt +Mt = Ptf(kt−1) + (1 + it−1)Bt−1 +Mt−1 + (1− δ)Ptkt−1 + Tt, ∀t ≥ 0
ψPtct ≤Mt−1 + Tt, ∀t ≥ 0
k−1 = k > 0, B−1 = B > 0, M−1 = M > 0
lim
t→∞
(Bt/Pt) = 0,
kt, ct,Mt ≥ 0, ∀t ≥ 0.
The notation is standard and is the same used in class: u(c) is the instantaneous utility
function; f(k) is the production function; Pt denotes the price of the good at date t; Bt is
the nominal amount of bonds the household chooses to hold at date t; Mt is the quantity
of money he chooses to carry from date t to date t + 1; it is the nominal interest rate
between dates t and t+ 1; ct is the household consumption at date t; kt is the amount of
capital he carries from date t to t + 1; Tt are cash transfers received at date t; β ∈ (0, 1)
is the discount factor and δ ∈ (0, 1) is the depreciation rate of capital. Define bt ≡ Bt/Pt,
mt ≡ Mt/Pt, πt ≡ (Pt − Pt−1)/Pt−1,τt ≡ Tt/Pt and let rt denote the real interest rate
between dates t and t+ 1.
The constraint ψPtct ≤ Mt−1 + Tt is interpreted as a standard cash-in-advance con-
straint, except that now only a fraction ψ ∈ (0, 1) of the household consumption is pur-
chased using cash.
Macroeconomía II, 2018/2 1
In what follows, assume that u(·) and f(·) satisfy all the usual assumptions that guar-
antee an unique interior solution (and thus the non-negativity constraints kt, ct,Mt ≥ 0
never bind in the optimal choice and you can ignore them). Moreover, assume the cash-
in-advance constraint binds at all dates in the optimal solution (which is true if it > 0,
for every t). There is no uncertainty and the household takes as given the path of all
exogenous variables.
1. Rewrite the budget constraint and the cash-in-advance constraint in real units (that
is, in terms of only the real variables {ct, kt, bt,mt, rt, τt}t≥0 and the inflation rate
{πt}t≥0). (Tip: use the Fisher equation to get rid of nominal interest rates).
2. Denote by {λt}t≥0 the Lagrange multipliers associated to the budget constraint and
by {µt}t≥0 the Lagrange multipliers associated to the cash-in-advance constraint.
Write down the Lagrangian using the budget constraints in real units and derive the
first order conditions for ct, kt, bt and mt.
3. In the optimal solution we get the following Euler equation:
u′(ct)
u′(ct+1)
= β (1 + rt)h(it−1, it, ψ)
where h(it−1, it, ψ) is a function of it−1, it and ψ. Derive the functional form of
h(it−1, it, ψ).
4. Interpret economically the Euler equation you found in the previous item when
ψ = 1. (Tip: you may find useful to rearrange the equation before your interpret
it.)
Exercise 2
Consider the CIA model in Section 3.3.1 of Walsh’s book (also seen in class) with one
modification: transfers received at date t can not be directly used to purchase goods at
date t, so that the the cash-in-advance constraint takes the form Ptct ≤ Mt−1. Assume
that u(c) = ln c. Also, assume that the CIA constraint always binds.
1. Write down the household’s decision problem.
2. Write down the household’s first-order conditions.
3. Derive the Euler equation.
Macroeconomía II, 2018/2 2
4. Show that:
Mt
Mt−1
= β (1 + it−1) .
5. Explain what happens to the nominal interest rate in 3 scenarios:
(a) Moneys grows at a constant rate: Mt/Mt−1 = 1 + µ;
(b) Money grows at an increasing rate: Mt/Mt−1 = 1 + µt;
(c) Money grows at a decreasing rate: Mt/Mt−1 = 1 + 1/ (µt).
Exercise 3
When we presented the CIA and MIU models in class, we wrote down the centralized
version of it: a single household decided how much to consume, how much capital, money
and bonds to hold. The household had access to a production function that would trans-
form this capital into consumption goods. Now we set up a decentralized version of the
CIA model.
Consider an economy in which there is a single consumption good that can be used as
capital or for consumption. Time is discrete and indexed by t ∈ {−1, 0, 1, 2, ...}. There is
a representative household and a competitive firm in this economy. Both the firm and the
household are price takers.
Household. The household owns the firm and chooses consumption and how much
capital, bonds and money to hold. At each date t, the household rents the capital it
brought from the previous period (kt−1) to the firm at a rental rate Pt × Rt (hence, the
rental rate in units of the consumption good is Rt). Every period, the household also
receives profits Dt from the firm. Hence, the household problem is:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t. Ptct + Ptkt +Bt +Mt = PtRtkt−1 + (1 + it−1)Bt−1 +Mt−1 + (1− δ)Ptkt−1 + Tt +Dt, ∀t ≥ 0
Ptct ≤Mt−1 + Tt, ∀t ≥ 0
B−1 = B > 0, M−1 = M > 0
lim
t→∞
(Bt/Pt) = 0,
ct,Mt ≥ 0, ∀t ≥ 0.
We are assuming that the profits Dt are only received at the end of the period (and
hence that money is not useful for transactions), while the transfers Tt are received at the
Macroeconomía II, 2018/2 3
beginning of the period (and hence can be used for transactions). That is why only Tt
shows up in the CIA constraint. The notation is standard and is the same used in class:
u(c) is the instantaneous utility function; Pt denotes the price of the good at date t; Bt is
the nominal amount of bonds the household chooses to hold at date t; Mt is the quantity
of money she chooses to carry from date t to date t + 1; it is the nominal interest rate
between dates t and t+ 1; ct is the household consumption at date t; Tt are cash transfers
received at date t;β ∈ (0, 1) is the discount factor and δ ∈ (0, 1) is the capital depreciation
rate. Define bt ≡ Bt/Pt, mt ≡Mt/Pt, πt ≡ (Pt−Pt−1)/Pt−1,τt ≡ Tt/Pt, dt ≡ Dt/Pt and let
rt denote the real interest rate between dates t and t+ 1. Household solve their problem
taking the sequence of prices, interest rates, transfers and profits as given.
Firm. The firm has access to a production function f(k̃). At each date t, the firm
chooses how much capital k̃t to rent from the household. Hence, at each date t the firm
problem is:
max
k̃t
Πt ≡ Ptf(k̃t)− PtRtk̃t
subject to k̃t ≥ 0.
Market clearing. The following market clearing must hold:
Bt = 0, ∀t ≥ 0
k̃t = kt−1, ∀t ≥ 0
ct + kt = f(kt−1), ∀t ≥ 0
Moreover, Tt = Mt−Mt−1, since the government can only make transfers by issuing more
money.
Assume that u(·) and f(·) satisfy all the usual assumptions that guarantee an unique
interior solution (and thus the non-negativity constraints never bind in the optimal choice
and you can ignore them). Moreover, assume the cash-in-advance constraint binds at all
dates in the optimal solution (which is true if it > 0, for every t). There is no uncertainty
and the household takes as given the path of prices, interest rates and all exogenous
variables.
1. Write down the household problem in real units.
2. Derive the first order conditions of the household problem.
Macroeconomía II, 2018/2 4
3. Write the first order condition of the firm.
4. Show that in equilibrium Rt = f ′ (kt−1) and rt = Rt+1 − δ .
5. Show that the steady state in the decentralized model is the same as in the centralized
model (that is, the model of section 3.3.1 in Walsh’s book, which is the one seen in
class).
Monetary equilibrium
Exercise 4
[Mankiw] In the country of Wiknam, the velocity of money is constant. Real GDP grows
by 3 percent per year, the money stock grows by 8 percent per year, and the nominal
interest rate is 9 percent. What is the growth rate of nominal GDP, the inflation rate and
the real interest rate?
Exercise 5
[Mankiw] Suppose a country has a money demand function Md/P = κy, where κ is a
constant parameter and y is real income.The money supply grows by 12 percent per year,
and real income grows by 4 percent per year.
1. What is the average inflation rate?
2. How would inflation be different if real income growth were higher? Explain.
3. How do you interpret the parameter κ? What is its relationship to the velocity of
money?
4. Suppose, instead of a constant money demand function, the velocity of money in
this economy was growing steadily because of financial innovation. How would that
affect the inflation rate? Explain.
Macroeconomía II, 2018/2 5
Proposed exercises
Exercise 6
Suppose the representative household chooses consumption of cash goods ({ct}t≥0), con-
sumption of credit goods ({dt}t≥0), nominal bond holdings ({Bt}t≥0) and hours of work
({nt}t≥0) to maximize
∞∑
t=0
βt [u(ct, dt)− γnt]
subject to the nominal budget constraint
Pt (ct + dt) +Mt +Bt = Wtnt + (1 + it−1)Bt−1 + Tt +Mt−1
and a cash-in-advance constraint of the form
Ptct ≤Mt−1 + Tt.
Where γ > 0, nominal wages are denoted byWt and the remaining notation is the standard
one. Denote the real wage by wt and real bond holdings by bt. The usual non-negativity
constraint on c and l must hold, as well as the solvency constraint (limt→∞ bt = 0).
Moreover, bonds and money at the beginning of period zero are given. Assume that:
u(ct, dt) = α ln ct + (1− α) ln dt, with α ∈ (0, 1).
1. Write the cash-in-advance constraint and budget constraint in real units.
2. Derive the first order conditions with respect to ct, nt, dt and bt.
3. Compute the steady state (assuming a constant inflation rate, real wage, consump-
tion, hours of work, bond and money holdings). Does this model exhibit superneu-
trality? Explain.
4. Maximizing utility at the steady state, what is the optimal inflation in this economy?
5. Write down an expression for the share of cash in total consumption as a function
of the nominal interest rate. How does the share of cash goods consumed depend on
the interest rate?
Macroeconomía II, 2018/2 6
Exercise 7
Consider the CIA model in Section 3.3.1 of Walsh’s book (also seen in class) with one
modification: to accumulate capital agents need to have cash in advance. In other words,
the CIA constraint now is given by:
Ptct + Pt (kt − (1− δ) kt−1) ≤Mt−1 + Tt
That is, now not only consumption must be purchased with cash, but also capital goods.
1. Present the Lagrangian of this problem, denoting by λt and µt the multipliers for
the budget constraint and the CIA constraint. Derive the first order conditions.
2. Show that:
u′(ct) = βu′(ct+1)
(
1
1 + it+1
)
f ′(kt) + βu′(ct+1)(1− δ)
3. Assuming a production function f(k) = kα, with α ∈ (0, 1) solve for the stationary
state capital kss. Is money superneutral in this version of the model? Give some
intuition on your result.
4. Now consider the same economy but with the usual CIA constraint Ptct ≤Mt−1 +Tt.
The steady state capital in that economy, with f(kt) = kαt , is given by:
kss =
 α
1
β
− 1 + δ
 11−α
Find the inflation rate that equates steady state capital in both economies and then
explain how this relates to Friedman rule.
Macroeconomía II, 2018/2 7
Macroeconomía II – Solutions
TA Session 3
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
Item 1
Dividing the budget constraint by Pt:
ct + kt + bt +mt = f(kt−1) + (1 + it−1)
Bt−1
Pt
+ Mt−1
Pt
+ (1− δ)kt−1 + τt
Rearranging:
ct + kt + bt +mt = f(kt−1) + (1 + it−1)
Bt−1
Pt−1
Pt−1
Pt
+ Mt−1
Pt−1
Pt−1
Pt
+ (1− δ)kt−1 + τt
ct + kt + bt +mt = f(kt−1) +
1 + it−1
1 + πt
bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt
And using the Fisher equation:
ct + kt + bt +mt = f(kt−1) + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt
Similarly, we divide the CIA constraint by Pt and rearrange:
ψct ≤
Mt−1
Pt
+ τt
ψct ≤
Mt−1
Pt−1
Pt−1
Pt
+ τt
ψct ≤
mt−1
1 + πt
+ τt
Macroeconomía II, 2018/2 1
Item 2
The lagrangian is given by
L =
∞∑
t=0
βt
{
u(ct)− µt
[
ψct −
mt−1
1 + πt
− τt
]
−λt
[
ct + kt + bt +mt − f(kt−1)− (1 + rt−1) bt−1 −
mt−1
1 + πt
− (1− δ)kt−1 − τt
]}
The FOC with respect to ct is:
βt [u′(ct)− λt − µtψ] = 0 (FOC1)
The FOC with respect to kt is:
−βtλt + βt+1λt+1 [f ′(kt) + 1− δ] = 0 (FOC2)
The FOC with respect to bt is:
−βtλt + βt+1λt+1 (1 + rt) = 0 (FOC3)
The FOC with respect to mt is:
−βtλt +
βt+1
1 + πt+1
(µt+1 + λt+1) = 0 (FOC4)
Item 3
Subtracting (FOC4) from (FOC3) and solving for µt+1:
βt+1λt+1 (1 + rt)−
βt+1
1 + πt+1
(µt+1 + λt+1) = 0
µt+1 = λt+1 [(1 + πt+1) (1 + rt)− 1]
Thus, replacing µt = λt [(1 + πt) (1 + rt−1)− 1] on (FOC1) we get:
u′(ct)− λt − ψλt [(1 + πt) (1 + rt−1)− 1] = 0
Using the Fisher equation it becomes:
Macroeconomía II, 2018/2 2
u′(ct) = ψλt
[
1 + it−1 +
1− ψ
ψ
]
(1)
Iterating (1) forward and dividing by (1):
u′(ct)
u′(ct+1)
= λt
λt+1
[
1 + it−1 + 1−ψψ
]
[
1 + it + 1−ψψ
]
But (FOC3) implies λt
λt+1
= β (1 + rt) and then:
u′(ct)
u′(ct+1)
= β (1 + rt)
1 + it−1 + 1−ψψ
1 + it + 1−ψψ
 (2)
Thus,
h(it−1, it, ψ) =
1 + it−1 + 1−ψψ
1 + it + 1−ψψ
.
Item 4
When ψ = 1 we have:
u′(ct)
1 + it−1
= β (1 + rt)u
′(ct+1)
1 + it
.
The LHS is the marginal benefit of one unit of consumption at date t divided by a function
of the opportunity cost of consuming at t (which is the foregone interest it−1 that the agent
has to incur by buying less bonds and carrying more money at t − 1). The RHS is the
marginal benefit of consuming at date t+ 1 divided by a function of the opportunity cost
of consuming at t + 1 (which is the foregone interest it that the agent has to incur by
buying less bonds and carrying more money at t). For the agent to be indifferent between
consuming at both dates, this “cost-benefit” ratio must be equal.
Exercise 2
Item 1
The problem is identical to the problem we had before, the only differencia is the CIA
constraint. The budget constraint in real terms remains the same, so I will skip the
derivation. Now we write the CIA constraint it as:
Ptct ≤Mt−1
Macroeconomía II, 2018/2 3
Which rearranging becomes:
ct ≤
Mt−1
Pt
= Mt−1
Pt−1
Pt−1
Pt
= mt−11 + πt
Hence, the household problem is:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t. ct + kt + bt +mt = f(kt−1) + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt, ∀t ≥ 0
ct ≤
mt−1
1 + πt
, ∀t ≥ 0
b−1 = b ≡ B/P−1 > 0, M−1 = m = M/P−1 > 0
lim
t→∞
(bt) = 0,
ct,mt, kt ≥ 0, ∀t ≥ 0.
where u(c) = ln c.
Item 2
The Lagrangian is given by
L =
∞∑
t=0
βt
{
u(ct)− µt
[
ct −
mt−1
1 + πt
]
−λt
[
ct + kt + bt +mt − f(kt−1)− (1 + rt−1) bt−1 −
mt−1
1 + πt
− (1− δ)kt−1 − τt
]}
The FOC with respect to ct is:
βt [u′(ct)− λt − µt] = 0 (FOC1)
The FOC with respect to kt is:
−βtλt + βt+1λt+1 [f ′(kt) + 1− δ] = 0 (FOC2)
The FOC with respect to bt is:
−βtλt + βt+1λt+1 (1 + rt) = 0 (FOC3)
Macroeconomía II, 2018/2 4
The FOC with respect to mt is:
−βtλt +
βt+1
1 + πt+1
(µt+1 + λt+1) = 0 (FOC4)
Item 3
We can rewrite (FOC1) and (FOC4) as (note that we used (FOC4) lagged):
u′(ct) = λt + µt
βλt−1
1 + πt
= λt + µt
Hence:
u′(ct) =
(1 + πt)λt−1
β
Dividing the equation above by the same equation forward and multiplying both sides by
1/β:
1
β
u′(ct)
u′(ct+1)
= 1 + πt1 + πt+1
λt−1
λtβ
But (FOC3) lagged immplies that λt−1
λtβ
= 1 + rt−1. Hence:
1
β
u′(ct)
u′(ct+1)
= 1 + πt1 + πt+1
(1 + rt−1)
Multiplying the RHS by (1 + rt)/(1 + rt):
1
β
u′(ct)
u′(ct+1)
= (1 + πt) (1 + rt−1)(1 + πt+1) (1 + rt)
(1 + rt)
Using the Fisher equation we get:
1
β
u′(ct)
u′(ct+1)
= 1 + it−11 + it
(1 + rt)
Since u′(c) = 1/c we can write:
ct+1 = β
1 + it−1
1 + it
(1 + rt) ct
Macroeconomía II, 2018/2 5
Item 4
Since the CIA constraint binds, we have that ct = mt−11+πt for all t. Plugging that in the
Euler equation we have:
mt
1 + πt+1
= β 1 + it−11 + it
(1 + rt)
mt−1
1 + πt
Mt
Pt
= β 1 + it−11 + it
1 + πt+1
1 + πt
(1 + rt)
Mt−1
Pt−1
Mt = β
1 + it−1
1 + it
1 + πt+1
��
��1 + πt
(1 + rt)Mt−1�����(1 + πt)
Mt = β
1 + it−1
1 + it
(1 + πt+1) (1 + rt)Mt−1
Using the Fisher equation:
Mt = β
1 + it−1
���1 + it �
���(1 + it)Mt−1
Mt
Mt−1
= β (1+ it−1)
Item 5
In scenario (a) we have that:
β (1 + it−1) = 1 + µ
Hence, interest rates remain constant. In scenario (b) we have that:
β (1 + it−1) = 1 + µt
Hence, interest rates increase as time passes. Finally, in scenario (c) we have that:
β (1 + it−1) = 1 + 1/ (µt)
Hence, interest rates decrease as time passes.
Note that in scenario (a) even though the money supply is increasing, the interest rates
remain constant. In scenario (b), the money supply is growing at a increasing rate, and
the interest are increasing. Note that this is the opposite one would expect from a simple
static model of money demand and money supply. This a common feature in this kind
of model. In a stochastic version of the CIA model, one can often find that interest rates
Macroeconomía II, 2018/2 6
increase after a random monetary expansion (this is called the liquidity puzzle).
Exercise 3
Item 1
Dividing the budget constraint by Pt:
ct + kt + bt +mt = Rtkt−1 + (1 + it−1)
Bt−1
Pt
+ Mt−1
Pt
+ (1− δ)kt−1 + τt +Dt
Rearranging:
ct + kt + bt +mt = Rtkt−1 + (1 + it−1)
Bt−1
Pt−1
Pt−1
Pt
+ Mt−1
Pt−1
Pt−1
Pt
+ (1− δ)kt−1 + τt + dt
ct + kt + bt +mt = Rtkt−1 +
1 + it−1
1 + πt
bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt + dt
And using the Fisher equation:
ct + kt + bt +mt = Rtkt−1 + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt + dt
Similarly, we divide the CIA constraint by Pt and rearrange:
ct ≤
Mt−1
Pt
+ τt
ct ≤
Mt−1
Pt−1
Pt−1
Pt
+ τt
ct ≤
mt−1
1 + πt
+ τt
Macroeconomía II, 2018/2 7
Hence, the household solves:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t. ct + kt + bt +mt = Rtkt−1 + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ (1− δ)kt−1 + τt + dt, ∀t ≥ 0
ct ≤
mt−1
1 + πt
+ τt, ∀t ≥ 0
b−1 = b ≡ B/P−1 > 0, M−1 = m = M/P−1 > 0
lim
t→∞
(bt) = 0,
ct,mt, kt ≥ 0, ∀t ≥ 0.
Item 2
The Lagrangian is given by
L =
∞∑
t=0
βt
{
u(ct)− µt
[
ct −
mt−1
1 + πt
− τt
]
−λt
[
ct + kt + bt +mt −Rtkt−1 − (1 + rt−1) bt−1 −
mt−1
1 + πt
− (1− δ)kt−1 − τt − dt
]}
The FOC with respect to ct is:
βt [u′(ct)− λt − µt] = 0 (FOC1)
The FOC with respect to kt is:
−βtλt + βt+1λt+1 [Rt+1 + 1− δ] = 0 (FOC2)
The FOC with respect to bt is:
−βtλt + βt+1λt+1 (1 + rt) = 0 (FOC3)
The FOC with respect to mt is:
−βtλt +
βt+1
1 + πt+1
(µt+1 + λt+1) = 0 (FOC4)
Macroeconomía II, 2018/2 8
Item 3
The first order condition of the firm is:
f ′
(
k̃t
)
= Rt (FOC5)
Item 4
Market clearing implies that k̃t = kt−1. Hence, (FOC5) implies that:
f ′ (kt−1) = Rt
Combining (FOC2) and (FOC3) we get:
βt+1λt+1 [Rt+1 + 1− δ] = βt+1λt+1 (1 + rt)
rt = Rt+1 − δ
Item 5
All the FOCs are the same as in the model seen in class, except for (FOC2). Plugging
f ′ (kt−1) = Rt in (FOC2):
−βtλt + βt+1λt+1 [f ′ (kt) + 1− δ] = 0 (FOC2’)
Note that (FOC1), (FOC2’), (FOC3) and (FOC4) are identical to the first-order conditions
in the centralized model seen in class. Hence, we should arrive at the same steady state,
since we are departing from the exact same set of equilibrium conditions.
Exercise 4
According to the quantitative equation (the notation is the same used in class):
MtV = Ptyt
Pt =
MtV
yt
Note that we did not add time subscripts to the velocity of money since it is constant.
Taking log’s we can write:
lnPt = lnMt + ln V − ln yt (1)
Macroeconomía II, 2018/2 9
lnPt−1 = lnMt−1 + ln V − ln yt−1 (2)
Subtracting (2) from (1):
lnPt − lnPt−1 = lnMt − lnMt−1 − (ln yt − ln yt−1)
Let ∆Xt ≡ Xt −Xt−1Since lnX − ln Y ≈ (X − Y ) /Y we can write:
πt =
∆Pt
Pt−1
≈ ∆Mt
Mt−1
− ∆yt
yt−1
Since ∆Mt
Mt−1
= 0.08 and ∆yt
yt−1
= 0.03 we have that inflation is πt ≈ 0.08 − 0.03 = 0.05. To
get the growth rate of nominal GDP (Yt) we depart from:
Yt = Ptyt
lnYt = lnPt + ln yt
Iterating it backwards and subtracting from the equation above we get:
∆Yt
Yt−1
≈ ∆Pt
Pt−1
+ ∆yt
yt−1
= πt +
∆yt
yt−1
Therefore, the growth rate of nominal GDP is 0.05 + 0.03 = 0.08 (alternatively, one could
use MtV = Ptyt = Yt to see that the growth rate of nominal GDP is equal to the growh
rate ofMt). Using the Fisher equation, we get a real interest rate of r = 0.09−0.05 = 0.04.
Exercise 5
Item 1
Let M denote the money supply. We can then write P = M/ (κy). Therefore, adding
time subscripts and taking the log’s:
lnPt = lnMt − ln κ− ln yt
Iterating it backwards and substracting from the equation we get the same relationship
we got in Exercise 3:
πt =
∆Pt
Pt−1
≈ ∆Mt
Mt−1
− ∆yt
yt−1
(3)
Therefore, since ∆Mt
Mt−1
= 0.12 and ∆yt
yt−1
= 0.04, we get that the inflation rate is 8% per year.
Macroeconomía II, 2018/2 10
Item 2
If the real grew at a higher rate, then inflation would be lower, as (3) makes clear. A
higher growth of real GDP increases the demand for real balances, which puts a positive
pressure on the real money supply, lowering prices (everything else constant).
Item 3
The parameter κ captures how sensitive is the demand for real balances to real GDP. In κ
is very large, any increase in real income implies a large increase in the demand for money.
Writing M 1
κ
= Py, one can see that 1/κ can be interpreted as the velocity of money. If
the velocity of money is very low, any increase in the amount of transactions people want
to make must be fulfilled by a large increase in the real money supply, since money does
not change hands a lot of times.
Item 4
Taking log’s from the quantitative equation we can write:
lnPt = lnMt + ln Vt − ln yt (4)
lnPt−1 = lnMt−1 + ln Vt−1 − ln yt−1 (5)
Subtracting 5 from 4:
lnPt − lnPt−1 = (lnMt − lnMt−1) + (ln Vt − ln Vt−1)−− (ln yt − ln yt−1)
Let ∆Xt ≡ Xt −Xt−1Since lnX − ln Y ≈ (X − Y ) /Y we can write:
πt =
∆Pt
Pt−1
≈ ∆Mt
Mt−1
+ ∆Vt
Vt−1
− ∆yt
yt−1
Hence, if ∆Vt > 0 because of financial innovation, we should expect the inflation rate to
increase. Intuitively, if the velocity of money is increasing, people demand less money for
their transactions (since the same dollar bill is being used a lot of times for transactions).
For a given nominal money supply, prices will have to increase to guarantee that the supply
of real balances falls, matching the lower demand.
Macroeconomía II, 2018/2 11
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